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<b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) ) ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) ) ) implies <font color="Maroon" title="c2">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:2_1_2"/><i><font color="Green" title="E3">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) )
 <b>and </b><br/><a NAME="E2:2_1_2"/><i><font color="Green" title="E4">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X2</font></span><br/>

<a NAME="E3:2_1_2"/><i><font color="Green" title="E5">A6</font></i>: 
<font color="Maroon" title="c2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> )</span><br/>

<b>assume </b><a NAME="E1:2_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font></span><br/>

<a NAME="E2:2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="txt" href="#E1:2_1_2"><i><font color="Green" title="E3">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:2_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 <b>by </b><i><a class="txt" href="#E2:2_1_2"><i><font color="Green" title="E4">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:2_1_2"/>
<font color="Maroon" title="c3">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> )</span><br/>

<b>assume </b><a NAME="E1:2_1_2_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font></span><br/>

<a NAME="E2:2_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="txt" href="#E2:2_1_2"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:2_1_2_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font>
 <b>by </b><i><a class="txt" href="#E1:2_1_2"><i><font color="Green" title="E3">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:2_1_2"/>
<font color="Maroon" title="c2">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X2</font>
 <b>by </b><i><a class="txt" href="#E3:2_1_2"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
